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102 4. Seismic Ray Tracing for Minimum Time Path The preceding procedure for a source in the second layer can be generalized to a source in a deeper layer, say, the jth layer. Once a trial angle & is chosen, we may use Snell’s law to compute successive incident angles to each overlying layer until the trial ray reaches the surface at point A, with the r) coordinate given by (4.102) & = q+ + 1 i=J-1 hi tan 8i where 8, = 4*. The incident angles are related by (4.103) sin Oi -- - sin for 1 5 i < j Vi Vi+l where Bi is the incident angle for the ith layer with velocity vi and thickness hi. With the proper substitutions, Eq. (4.102) can also be used to calculate Al and A2. In this iterative procedure, the trial angle $* converges rapidly to the angle 4 whose associated ray path reaches the station within the error limit E. Since this ray path consists ofj segments of a straight line in each of the j layers, we can sum up the travel time in each layer to obtain the travel time from the source to the station. Knowing how to compute travel time for both the direct and the refracted paths, we can then select the minimum travel time path. The spatial derivatives and the take-off angle can be computed from the direction cosines using results from Section 4.4.1 as follows. Let us consider an earthquake source at point A with spatial coordinates (xA, y,, zA), and a station with spatial coordinates (xB, yB, zB). Let us choose a coordinate system such that points A and B lie on the qz plane, and A’ is the projection of A on z = zE. In Fig. 25, let us consider an element of ray path ds with direction angles a, p, and y, and components dx, dy, and dz (with respect to the x, y, and z axes, respectively). In Fig. 25a, the projection of ds on the r) axis is sin y ds. In Fig. 25b, this projected element can be related to the components dx and dy of ds by (4.104) where a’ and p’ are the angles between the r) axis and the -x and y axes, respectively. Consequently, from Eq. (4.104) and the definition of direction cosine for y, we have (4.105) dx = cos a’ sin y ds, dx/ds = cos a’ sin y, dz/ds = cos y dy = cos p’ sin y ds dy/ds = cos p‘ sin y The angles a’ and p‘ can be determined from Fig. 25b as
4.4. Computing Travel Time and Derivatives 103 (sin y ds) L I , I I I .. I Fig. 25. Diagram of the projections of an element of ray path ds on theqz plane (a) and on the xy plane (b). (4.106) cos (Y' = (xB - xA)/A, COS~' = (YB - yA)/A whereA is the epicentral distance between points A' andB and is given by (4.107) A = [(XB - Xd)' -l (YE- YA)']''~ For the refracted waves in a multilayer model (see Fig. 23), the direction angle y at the source is the take-off angle for the refracted path. If the earthquake source is in thejth layer and the ray is refracted along the top of the kth layer, then by Eq. (4.96)
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PRINCIPLES AND APPLICATIONS OF MICR
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Contents FOREWORD PREFACE vii ix 1.
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Foreword Earthquakes occur over a c
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X PI-
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2 1 . It1 trodir ctivn of the earth
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frequency of occurrence N by (I .2)
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6 1. Introduction the initial P-ons
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8 1. Ititrodiictioti that changes i
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10 1. Introduction Fig. la. Schemat
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12 Fig. b. Map showing seismographi
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14 1. Introduction indicated the la
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~ 16 2. Ins trumeri tic t ion Syste
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18 2. Itistrirmeritation Systems In
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20 2. Ins trumeti ta tim S ys tems
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22 2. Ins tru inen ta t ion Systems
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24 2. Iris trumentrt tion Systems F
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26 2. Itistrutnetztntioti System 5H
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28 2. Instrumentntior? Systems Fig.
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30 2. Instrumentation Systems The o
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32 2. Instrumentation Systems other
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34 2. Instrumentation Systems 1 0 7
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a,No ak) ak) ak) TABLE I Seismic Sy
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38 2. Znstrumentution Systems FREQU
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40 2. Instrumentatiori Systems mome
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42 2. Ins tru rneti totion Sys tenz
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44 2. Instrumentation Systems magne
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3. Data Processing Procedures Micro
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48 3. Datu Processirig Procedures c
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50 3. Data Processitig Procedures c
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Page 90 and 91: 4. Seismic Ray Tracing for Minimum
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Page 120 and 121: 106 5. Inversion and Optimization u
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Page 132 and 133: 118 5. In versiori nnd Optimization
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Page 144 and 145: 130 6. Methods of Data Analysk 6.1.
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Page 156 and 157: 142 6. Methods of Data Analysis (a)
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152 6. Methods of Data Analysis (6.
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154 6. Methods of Data Analysis qua
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156 6. Methods of Datu Analysls mag
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158 6. Methods of Data Analysis 6.5
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160 6. Methods of Data Analysis (6.
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162 6. Methods of Data Analysis fro
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7. General Applications Data from m
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166 7. General Applications in ampl
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168 7. General Applications Rangely
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170 7. General Applications The inv
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8.1. Seistnicit~ Pritterns 199 made
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8.1. Seismicity Patterns 201 only p
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8.1. Seismicit! Plitterrzs 203 betw
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Keilis-Borok et al. (1980b) suggest
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8.1. Srismic.itJ Ptrtterns 207 of t
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8.2. Temporal V'iricrtions of Seism
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8.3. Temporiil Vuriiitiotis qf Otli
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8.3. TemporuI Variations of Other S
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8.3. Temporal Vcrridoris of‘ Othe
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8.3. Totnportrl Vtrritrtiotis of' O
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9. Discussion Microearthquake studi
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Automation usually does not reduce
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Glossar? of' A bbreiqiations 227 da
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Glossary of AbbrciTintions 229 said
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References Acton, F. S. (1970). "Nu
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References 23 3 Asada, T. ( 1957).
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Rtjfcrcric-es 235 travel-time chang
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Refererices 237 Cleary, J. R., Wrig
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References Eaton, J. P. (1976). Tes
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References 24 1 Francis, T. J. G.,
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243 Hagiwarii. T., and Iwata. T. (1
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R efereiices 245 Horner, R. B., Ste
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References 247 Kagan, Y. Y., and Kn
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Referetices 249 Kodama, K. P., and
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25 1 Lin, M. T., Tsai, Y. B., and F
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References 253 Mikumo, T., Otsuka,
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R eferetices 255 Nordquist, J. M. (
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Re feret ices 257 F'rothero, W. A.
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259 Sauber. J., and Talwani, P. (19
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References 26 1 Stephens, C. D., La
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Keferetices 263 Tsujiura, M. (1978)
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R ef ere1 ices croearthquake observ
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